Consider a countable group G acting on the unit sphere S in the space of L² functions on G by the regular representation. Given a homology class h in the quotient space S/G, one defines the spherical Plateau solutions for h as the intrinsic flat limits of volume minimizing sequences of cycles representing h. Interestingly in
some special cases, for example when G is the fundamental group of a closed hyperbolic manifold of dimension at least 3, the spherical Plateau solutions are essentially unique and can be identified. However in general not much is known. I will discuss the questions of existence and structure of non-trivial Plateau solutions. I will also explain how uniqueness of spherical Plateau solutions for hyperbolic manifolds of dimension at least 3 implies stability for the volume entropy inequality of Besson-Courtois-Gallot.